Comparison of sensitivity among dynamic balance measures during walking with different tasks

Although various measures have been proposed to evaluate dynamic balance during walking, it is currently unclear which measures are most sensitive to dynamic balance. We aimed to investigate which dynamic balance measure is most sensitive to detecting differences in dynamic balance during walking across various gait parameters, including short- and long-term Lyapunov exponents (λs and λl), margin of stability (MOS), distance between the desired and measured centre of pressure (dCOP–mCOP) and whole-body angular momentum (WBAM). A total of 10 healthy young adults were asked to walk on a treadmill under three different conditions (normal walking, dual-task walking with a Stroop task as an unstable walking condition, and arm-restricted walking with arms restricted in front of the chest as another unstable walking condition) that were expected to have different dynamic balance properties. Overall, we found that λs of the centre of mass velocity, λs of the trunk velocity, λs of the hip joint angle, and the magnitude of the mediolateral dCOP–mCOP at heel contact can identify differences between tasks with a high sensitivity. Our findings provide new insights into the selection of sensitive dynamic balance measures during human walking.


Introduction
Human postural balance can be defined as the capability of maintaining an upright posture under the effect of perturbation due to external or internal factors, coming respectively from the environment or from the human body itself [1].Human balance can be studied in static (e.g. during standing still) or dynamic (e.g. during walking) conditions.Our focus here will remain on dynamic balance and the associated dynamic balance measures.Various dynamic balance measures have been proposed to evaluate the risk of falling, and their effectiveness and validity have recently been compared in literature reviews [2,3].Hamacher et al. [2] performed a comparative effect size (Cohen's d) analysis on the variability and dynamic stability of foot trajectories during level walking among young and elderly individuals based on 29 studies.The aforementioned systematic review identified balance measures that can discriminate between elderly and young individuals and between those more and less prone to falls among elderly individuals.Another study by Bruijn et al. [3] reviewed the nine currently available balance measures and assessed their validity, particularly in terms of construct validity, predictive validity in simple models, convergent validity in experimental studies and predictive validity in observational studies.We note that both studies are meta-analyses based on previous literature.
Various dynamic balance measures have been used to assess balance experimentally in healthy young adults, elderly individuals and individuals with gait disorders [4][5][6][7][8][9][10][11][12].This research primarily focused on the dynamic balance under the effect of perturbation due to the internal factors, which are different from the external factors such as a slip on a slippery floor.Most of them have measured and compared multiple balance measures for different groups simultaneously within the same experimental conditions, but few have discussed which measure can be most sensitive to discriminate between stable and unstable gait.Therefore, it is still unclear which balance measure is most sensitive to identifying dynamic balance during walking, and the purpose of this study was to investigate which balance measures are more sensitive to discriminating stable and unstable gait.
Gait variability, defined as stride-to-stride fluctuations while walking, has been closely associated with falls.Hausdorff et al. [14] reported that elderly individuals who had experienced a fall had a more unstable gait with larger variability in stride time and swing time than did those who had not fallen.The Lyapunov exponent is a measure that quantifies the degree of separation of two very close trajectories in state space and the local divergence of neighbouring trajectories in state space reconstructed from kinematic data [19], with large Lyapunov exponent values indicating a less stable system.A short-term Lyapunov exponent (λ s ) is the initial phase of divergence, whereas a long-term Lyapunov exponent (λ l ) is the terminal phase of divergence.Kang & Dingwell [19] indicated that elderly individuals had a larger λ s of the trunk and pelvis than did younger ones, indicating that the trunk motion best represents the age-related differences in dynamic balance during walking.MOS is the distance between the extrapolated centre of mass of the whole-body (XCOM body ) and the border of the base of support (BOS) [22].XCOM body is defined as the vector sum of the whole-body centre of mass (COM body ) position and a proportion of its velocity.It is evaluated as stable if the projection point of XCOM body is captured by the BOS and unstable if it is outside the BOS.Therefore, a larger MOS allows for greater system tolerance against perturbation in both anteroposterior (AP) and mediolateral (ML) directions [23].dCOP is the virtual centre of pressure at which the sum of the moments acting on the COM body during walking becomes zero when the actual centre of pressure (i.e.mCOP) coincides with the dCOP [13].Therefore, the dCOP-mCOP is proportional to the magnitude of the moment acting on the COM body , and an increase in dCOP-mCOP results in an increase in the external moment around the COM body , potentially leading to the loss of postural balance.Yamaguchi et al. [13], who investigated dCOP-mCOP in a turning gait on a slippery floor, found that dCOP-mCOP in the lateral direction was a good predictor of falls.The WBAM is the sum royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 230883 of the angular momentum around the COM body and the angular momentum of body segments [4,28,29].In cases where the WBAM increases due to complex movements or where a decrease in the ability to regulate the WBAM exists, a higher level of balance control is required and the possibility of falls increases [28].Vistamehr et al. [28] compared the WBAM between healthy adults and participants who had experienced a stroke and suffered from hemiplegia during straight walking and climbing steps.Notably, they found that individuals who had experienced a stroke had a larger frontal WBAM range.
By comparing these dynamic balance measures, we aimed to investigate which measures are most sensitive in detecting differences in dynamic balance between the following three walking conditions: normal walking, dual-task walking with a Stroop task as an unstable walking condition, and armrestricted walking with arms restricted in front of the chest as another unstable walking condition.Arm swing during walking causes changes in WBAM [30] and walking speed [31].It is also reported that the Lyapunov exponent is affected due to arm constraint [32].Therefore, we hypothesized that arm-restricted walking is thought to affect biomechanical balance measures such as WBAM, MOS and dCOP-mCOP as well as balance measures related to local stability such as the Lyapunov exponent.On the other hand, it has been pointed out that dual-task walking during the Stroop test causes changes in trunk movement variability and Lyapunov exponent [33].These findings suggest that traditional gait variability and Lyapunov exponent will be affected during dual-task walking, which is our second hypothesis.Since all the above findings were acquired in different experimental studies and have never been evaluated under same experimental conditions, we aimed to verify these hypotheses within the same controlled experimental paradigm.The abbreviations and symbols used throughout the paper are summarized in table 1.

Participants
This study included 10 healthy young adult males whose age, height and body mass were 28.8 ± 6.0 years, 1.70 ± 0.06 m and 66.9 ± 7.2 kg (mean ± standard deviation), respectively.The participants had no history of neuromuscular, cognitive, musculoskeletal disorders or injuries that could affect their balance and walking ability.All participants were informed of the protocol and provided written informed consent prior to experiment.The protocol was approved by the ethics committee at the University of Tokyo.

Experimental procedure
The experimental set-up used in this study consists of a three-dimensional motion analysis system with seven infrared cameras (Qualisys Motion Capture Systems, Qualisys AB), an instrumented treadmill with two force plates embedded in each side (Instrumented treadmill FIT, Bertec Corporation) and a personal computer for data measurement.Infrared reflective markers were attached to 20 major joints throughout the participant's body (i.e. the ear, shoulder, elbow, wrist, anterior superior iliac spine (ASIS), trochanter, knee, ankle, fifth metatarsal and heel of each side), from which three-dimensional motion data were obtained.In addition, each force plate measured the ground reaction force (GRF) and mCOP independently for the left and right feet.The sampling frequencies for GRFs and three-dimensional motion data were 1 kHz and 200 Hz, respectively.
Participants were asked to walk on the treadmill for 10 min under three walking conditions: (i) normal walking, (ii) dual-task walking, and (iii) arm-restricted walking.The treadmill speed was set to 1.0 m s −1 .During normal walking, participants were instructed to walk at their preferred stride length and cadence.During dual-task walking, participants walked while performing a cognitive task called the Stroop test [34], in which they answered aloud the colour of the letters displayed on a monitor set in front of them at eye level while walking.The letters were displayed on the monitor in intervals of 3 to 5 s.During arm-restricted walking, the participant walked with their arms crossed in front of their chest.Each condition was performed once at random order.To avoid fatigue effects, at least a 10 min break was provided between trials.

Data analysis
The total number of steps during a 10 min trial in each task was about 1300 steps regardless of the participant.Stationary and non-missing data were analysed using 450 successive strides (900 steps) in royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 230883 Kinetic and kinematic data were low-pass filtered using a fourth-order Butterworth filter with zero lag and cut-off frequencies of 10 Hz.The position of the whole-body COM was estimated using a 12-segment model involving the motion data.Heel contact and toe-off were determined based on vertical GRF (greater than 50 N for heel contact and less than 50 N for toe-off, respectively).

Gait variability
Stride time (ST), single-leg-stance duration (D SLS ), double-leg-stance duration (D DLS ), stride length (SL) and step width (SW) were calculated using the heel contact and toe-off.ST was defined as the duration between two consecutive heel contacts of the same foot.D SLS was defined as the duration between the toe-off and heel contact of the same foot.D DLS was defined as the duration during double stance.SL was defined as the total length of two steps during a stride.SL was defined as the anteroposterior (AP) distance between the heel markers at heel contact.SW was defined as the ML distance between heel markers at heel contact.The mean value of the coefficient of variation (CV) of each variable was calculated.

Lyapunov exponent
The three-dimensional time series of COM body velocity (vCOM body ), COM trunk velocity (vCOM trunk ), COM thigh velocity (vCOM thigh ), COM shank velocity (vCOM shank ) and COM foot velocity (vCOM foot ) were used to calculate the Lyapunov exponent.The one-dimensional time series of the hip joint angle (θ y hip joint ), knee joint angle (θ y knee joint ) and ankle joint angle (θ y ankle joint ) in the sagittal plane were also used.Data were resampled so that the number of data points per SL was 100 [35,36].For each threedimensional time series of each COM velocity data, a nine-dimensional state space S(t) was defined using time-delayed copies and embedding dimension, ð2:1Þ where v x , v y and v z are the velocities in the x, y and z axes, while t and t are the time and time delay, respectively.For each one-dimensional time series of each lower limb joint angle, a six-dimensional where u y is the joint angle in sagittal plane.Time delays were calculated using the first minimum point of the average mutual information, and embedding dimensions were calculated using the false nearest neighbour method [37].For each time series, the first minimum point of the average mutual information was calculated for each subject, gait condition and axis, and then averaged to obtain the time delay.The divergence curve was calculated using the algorithm proposed by Rosenstein et al. [38].λ s was calculated as the slope between 0 and 0.5 strides, whereas λ l was calculated as the slope between 4 and 10 strides in the divergence curve [3,18].

Margin of stability
The XCOM body was defined as the vector sum of the COM body position and a proportion of its velocity as follows [22]: where, r XCOM body is the XCOM body position, r COM body is the COM body position, v COM body is the COM body velocity, g is the gravity constant, and l is the distance between the COM body position and ankle marker.The MOS was calculated as the distance between XCOM and the border of BOS as follows [22]: where, r border is the position of the border of BOS.
In this study, the MOS at the heel contact and the minimum MOS during single-leg stance were calculated [22][23][24][25]27].ML MOS at heel contact (ML MOS HC ) was defined as the distance in the ML direction between the XCOM body and the heel marker position at heel contact.The minimum ML MOS during single-leg stance (ML MOS min ) was defined as the minimum distance between the XCOM body and the fifth metatarsal marker position during single-leg stance in the ML direction.AP MOS at heel contact (AP MOS) was defined as the AP distance between the XCOM body and the heel marker position of the leading foot at heel contact.As to AP direction, only the MOS at heel contact was calculated given that the minimum AP MOS during single-leg stance agrees with that at heel contact.The mean value of magnitude normalized to the participant's body height was used for comparison.

Distance between the desired and measured centre of pressure
The dCOP position (x dCOP , y dCOP ) was calculated from the rotational motion equation of the inverted pendulum consisting of COM body and mCOP.From the equation of rotational motion of the inverted pendulum in the frontal plane (x-z plane) and sagittal plane ( y-z plane), the following equations were obtained [13]: and where I y and I x are the moment of inertia in the frontal plane and sagittal plane; € u y and € u x are the angular acceleration around COM body in the frontal plane and sagittal plane; and x COM body , y COM body and z COM body are the position of COM body in the x, y, and z directions, respectively.For equations (2.5) and (2.6), when € u x , € u y ¼ 0, the moment acting around COM body becomes zero.Then, let x mCOP ¼ x dCOP , y mCOP ¼ y dCOP , the dCOP position was calculated by the following equations: The distance between dCOP and mCOP (i.e.dCOP-mCOP) throughout the whole gait cycle was calculated.The root mean square (RMS) during a single stride, range (max minus minimum) during the stride and RMS dCOP-mCOP during the single stance phase were calculated.We found that there were three and four unique peaks characterizing the dCOP-mCOP dynamics for ML and AP directions during single stance phase as shown in figure 1.
The mean value of each peak dCOP-mCOP normalized to the participant's body height was calculated and used for comparison.

Whole-body angular momentum
WBAM was calculated using the following equation [28]: where, r COM i , v COM i , m i , I i and v i are the COM position, COM velocity, mass, moment of inertia and angular velocity of ith body segment, respectively.The RMS of the WBAM during the single stride, range (maximum−minimum) of the WBAM during the single stride and the range of WBAM during the single stance phase were calculated.The mean value of each WBAM normalized by the product of the participant's body height, body weight and ffiffiffiffiffiffi ffi g=l p were calculated for comparison.

Statistical analysis
We performed a paired t-test with Bonferroni correction to determine significant differences in balance measures between the three different walking tasks.The significant level was set at 0.05/3.The effect size for t-tests was also reported using Cohen's d, with values of 0.2-0.4,0.4-0.8 and greater than 0.8 indicating small, moderate and large effects, respectively [39].Receiver operating characteristic (ROC) analysis was also performed, after which the area under the curve (AUC) was calculated and used to investigate the discriminatory power of each balance measure between the normal walking task and the unstable walking tasks (i.e.dual-task and arm-restricted walking).An AUC between 0.5 and 0.7, between 0.7 and 0.9 and above 0.9 indicates low, medium and high discriminatory power, respectively [40].Paired t-test and effect size analyses were performed using Microsoft Excel (Microsoft, Redmond, WA, USA).Effect size and ROC analyses were performed using Matlab version 8.3 (Mathworks, Natick, MA, USA).

Results
Table 2 summarizes the mean value of each balance measure for the three gait tasks and the results of the t-tests, effect sizes (Cohen's d values) and AUC values between normal, and dual-task or arm-restricted walking.
The CV values of the ST, D SLS and SL during arm-restricted walking were significantly larger than those during normal walking ( ps < 0.05).The AUC values of these measures were found to discriminate between normal and arm-restricted walking with moderate discriminatory power (0.7 < AUC < 0.9).By contrast, no  the current study.This could be attributed to the possibility that the cognitive load of the Stroop test used in this study was insufficient for the participant population, and the memory required for processing the walking motion could be handled without interference.Thus, a larger cognitive load may have resulted in a significant difference in the balance measures.The results partially support our second hypothesis, in which dual-task walking during the Stroop test could affect the Lyapunov exponent but not the biomechanical balance measures.The relatively high sensitivity of λ s of θ y hip joint , λ s of vCOM body and λ s of vCOM trunk may be explained by the fact that the trunk (50%) and upper leg (20%) occupy the majority of body mass that primarily affects dynamic balance.Maintaining trunk stability is suggested to be one of the most important aspects of dynamic balance [41,42].Given that the hip joint connects the trunk and upper leg, its behaviours can also be critical in dynamic balance.Hamacher et al. [2] indicated that during level walking, the λ s using trunk motion data can detect differences in local dynamic stability between older and younger populations with higher sensitivity, which supports our results.
Our results indicated that λ s can detect gait instability with higher sensitivity than gait variability parameters and balance measures derived from biomechanics, such as MOS, WBAM and dCOP-mCOP.In fact, Bruijn et al. [3] confirmed the validity of gait variability measures and λ s for estimating gait stability during gait with a small perturbation without having large external mechanical perturbations.In the current study, small perturbations were continuously presented during walking with the unstable walking tasks.Gait variability measures (i.e.CV values of gait parameters) only quantify the average differences between strides, independent of the temporal order in which strides occur, indicating that such measures contain no information about how the locomotion system responds to perturbations either within or across strides.However, λ s quantifies how fast gait patterns diverge after infinitesimal perturbation, which should exhibit a higher sensitivity compared with the gait variability measures used in the current study.This is also confirmed by Dingwell et al. [5], who showed that stride-to-stride variability and local dynamics stability quantify fundamentally different aspects of locomotor behaviour and that gait variability measures poorly predict local stability.
Our results also suggests that the ML Peak HC of dCOP-mCOP was able to identify differences in dynamic balance between normal and unstable walking.The derivative of WBAM is proportional to the dCOP-mCOP given that the derivative of WBAM is the moment around COM body , which is proportional to the dCOP-mCOP.Therefore, a rapid change in WBAM will appear as a peak in the dCOP-mCOP.As shown in figure 2, we performed multiple comparisons between normal and armrestricted walking using the Benjamini-Hochberg method (BH method) [43] by 1% on the time series data of frontal WBAM normalized to the 0%-100% gait cycle.The figure shows a statistically significant difference ( p < 0.05) at 39%-45% and 88%-95% (around heel contact).Similarly, as shown in figure 3, multiple comparisons were made using the BH method for the ML dCOP-mCOP between normal and arm-restricted walking.In the ML dCOP-mCOP, a statistically significant difference ( p < 0.05) was observed at 42%-51% and 93%-100% (including the heel-ground contact).In the unstable phase before heel contact, the WBAM changed constantly in normal walking, whereas the absolute value of the WBAM during arm-restricted walking increased rapidly.In the phase just before heel contact, it would be difficult to control the WBAM during arm-restricted walking, and the moment around the COM body increased.MOS is theoretically regulated by step length and SW in the AP and ML directions, respectively [22,23].We confirmed that the mean values of SL and SW did not significantly differ among gait tasks ( p > 0.05), which could be attributed to the miniscule perturbations in each unstable walking condition.This could be the reason why MOS could not identify a difference in the dynamic balance between the normal and unstable walking tasks performed herein.

Study limitations
Some limitations of the current study should be considered.First, our study was conducted on a treadmill; hence, it is unclear whether our results can be applied to overground walking.Second, because the arm-restricted walking condition can be a highly unstable gait configuration, the results obtained in this study may not be applied to unstable gait in general.Another limitation was that the participants were all young adult males.Therefore, whether our results could be applied to dynamic balance alterations associated with disability or ageing needs to be further investigated.Although we investigated effect size, the small sample size (n = 10) could limit our results.As such, further studies with a larger sample size are needed.While our reflective marker placement is sufficient for extracting the parameters analysed in the current study, future work should consider if the observations demonstrated herein would work with standard marker placement (e.g.Helen Hayes marker set).

Conclusion
This study has been an attempt to compare dynamic balance measures, such as gait variability measures, Lyapunov exponents, margin of stability, desired centre of pressure and whole-body angular momentum, through identical gait trials with different gait tasks (normal, dual-task and arm-restricted walking).The findings of the current study indicate that λ s of vCOM body , λ s of vCOM trunk , λ s of θ y hip joint and the magnitude of the ML Peak HC of dCOP-mCOP are dynamic balance measures with high sensitivity for detecting difference in dynamic balance between normal walking and unstable walking tasks.Our findings provide new insights into the selection of more sensitive dynamic balance measures and relationships among dynamics balance measures.

Figure 2 .
Figure 2. Comparison of the WBAM in frontal plane between normal walking and arm-restricted walking.

Figure 3 .
Figure 3.Comparison of the dCOP-mCOP in the ML direction between normal and arm-restricted walking.
x , F y , F z ) H vector of whole-body angular momentum I moment of inertia of inverted pendulum consisting of whole-body centre of mass and measured centre of pressure, coordinates (I x , I y ) I i moment of inertia of ith body segment l distance between the whole-body centre of mass and ankle x , v y , v z ) (Continued.) royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 230883 the middle of each trial.Matlab (Mathworks, Natick, MA, USA) was used for subsequent analyses.

Table 1 .
(Continued.)joint angle in the sagittal plane θ y ankle joint ankle joint angle in the sagittal plane θ y hip joint hip joint angle in the sagittal plane θ y knee joint knee joint angle in the sagittal plane v i angular velocity vector of centre of mass of ith body segment royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 230883 state space was defined using time-delayed copies and embedding dimension,

Table 2 .
Figure 1.Temporal change in dCOP-mCOP during a gait cycle.RHC, right heel contact; LHC, left heel contact.P i_R , and P i_L represent ith peak dCOP-mCOP in stance phase for right and left feet.(a) ML direction.(b) AP direction.royalsocietypublishing.org/journal/rsos R. Soc.Open Sci.11: 230883 Mean values of each balance measure for the three gait tasks and the results of the t-tests, effect sizes (Cohen's d values) and AUC values between normal and dual-task or arm-restricted walking.